It's known that $S^2$ has a unique complex structure, which can be proved using the Riemann-Roch theorem.
Is there an elementary proof without using the Riemann-Roch theorem?
By an "elementary" proof, I mean a proof using mainly techniques from complex analysis. In other words, suppose we were in the early to mid of 19th century (before 1865, so the Riemann-Roch theorem was not proved yet). So we didn't know future concepts such as divisors, line bundles, cohomology and even didn't know formal definitions of groups and rings. But we did have plenty knowledge of complex analysis.
Suppose we knew how to define a complex manifold and we believed $S^2$ has a unique complex structure. How would we prove it?
You are asking about a special case of the Uniformization Theorem, namely, for surfaces homeomorphic to $S^2$: If a Riemann surface is homeomorphic to $S^2$ then it is conformally isomorphic (biholomorphic) to $S^2$, the Riemann sphere. A caveat:
There was no satisfactory definition of Riemann surfaces before 1912. Accordingly, none of the early proofs (including the RR Theorem) were rigorous.
Now, to answer your question:
If you are asking about a proof of the above special case of the Uniformization Theorem (UT) which uses tools similar to the ones taught in Complex Analysis classes, e.g. to the proof of the Riemann Mapping Theorem, and avoiding the potential theory and the RR Theorem, you can find one in Caratheodory's book "Conformal Representation", 1932.
Caratheodory also gives a proof of the full UT using normal families, etc, but in a different book (I forgot the title). If you are interested, I can find a reference.